Help me out with thinking this through; I’m unsure about what I suggest below and would like to figure it out.

(*To commenters:* I’m particularly interested in how to model the strategic behavior here – I’ve suggested that it could be thought of by reference to stock portfolio diversification theory, but as several have pointed out, that’s not quite right, partly because diversification in the stock market starts from the assumption that you have a hundred dollars and you can buy one share at a dollar each of 100 stocks or 100 shares of one stock; in the case of college applications, you can’t submit more than one application to a school. That disanalogy seems to me to point even more strongly why you would want to apply to a lot of schools especially if others were doing so, but I’m interested in understanding the abstract gaming behavior here. I’d be particularly interested to know if this situation is actually modeled in the game theory literature or if it matches up to some kind of standard game – I went through one of my game theory texts but didn’t see exactly the game I was looking for. But I’m quite interested – less in the practicalities, frankly, than the strategic behavior model in the abstract.)

(*PS to commenters:* Alternatively, here’s another way to frame the question. If I were trying to convince the guidance office on a purely rational choice basis, per the hypothetical below, that the high school was hurting student’s chances of getting into the colleges of their choice by limiting applications in this way, what would the abstract rational choice argument be? Would it include proposition one below, or would it include some corrected version of proposition two, and how would you state that in the abstract?)

Assume that you are a student applying to college. Assume also that your high school has a declared policy limiting the number of schools to which you can apply. It says it does so in order that you will “take applying seriously.”

You rather suspect that the school does so in order to be able to push particular students it favors for particular colleges, so that too many students do not create too much “noise” and competition for a place that the high school counselor thinks is most competitively filled by a particular student. The high school’s incentive, in other words, is first to maximize its ability to get students into particular colleges as an institution seeing the students as a school cohort, not to simply support each student in his or her efforts on an individual basis. One effect of this is to favor students who come from wealthy, powerful, or highly connected legacy families, since those students are most likely in the first place to be able to get into the most competitive schools. But although you suspect this, it is rather difficult to prove without something like discovery in a lawsuit to find out exactly on whose behalf the counselors call certain colleges and discourage other students from applying at all. (But let’s leave all that aside to focus on a more generalized strategic question; I’ll put up a post on Richard Kahlenberg’s new book on legacy admissions later.)

*Proposition one*. If we assume that the student has more than a zero chance of getting into a top institution – not a slam dunk, but not ruled out altogether, or at a minimum uncertain – in an environment with many students applying, then simply as a matter of increasing the student’s odds, the student would do better to apply, say, to all the top colleges merely as a matter of odds. This is all things equal better for the individual student even if it makes it more difficult for the high school to favor some students over others in applications to particular colleges.

*Proposition two* – and this is the one that particularly interests me as to whether it is true or not. Assume that this particular high school limits the number to which one can apply to six. If the trend is for students at other high schools across the country to apply to a much larger number of colleges, then this high school student does not just suffer from missing the odds of applying to a greater number of colleges – this student is penalized for *not* having diversified applications across colleges relative to other applicants.

It seems to me – maybe – akin to the condition in portfolio theory that market prices in shares assume that risk that can be eliminated through diversification will be eliminated that way and will not be compensated for in share prices; the market assumes diversification. In college admissions, if large numbers of students diversify their risk by applying to a large number of colleges – diversifying their holdings of lottery tickets to those colleges – a student who fails to to diversify by applying to a large number of colleges will be penalized with lower odds of admission even beyond simply increasing the odds, other things equal.

Is this second proposition correct or is it not correct, or does proposition two merely collapse into proposition one? If there were only one student in the world applying, and colleges were not required to take any students at all, and if the chances of admission to any one college were genuinely unknown and uncertain as to the student, then presumably the student should increase the odds and hedge against the uncertainty by applying to more rather than fewer schools. That is proposition one.

The second proposition adds other students also applying. Given that they are also bidding for entry, and assuming the same unknowns, they will have the same reasons to increase the number of colleges to which they apply. So everyone ideally wants to apply everywhere. But if for some reason everyone else automatically applies to 100 colleges and you apply to only one, are your chances *worse* because of the fact that everyone else *has* applied to every available college? That is proposition two – the addition of new players to the game who hold in effect a diversified portfolio while you hold only one application, and unlike stock, you haven’t bought lots and lots of application lottery tickets to that college; you can only buy one.

My point is two fold, if I’m right. One is that there are two very good reasons, one intrinsic and the other strategic, for increasing without limit the number of colleges to which you apply. The other is that if you fail to do so when others do, you put yourself in a distinctly worse relative position. Is this akin to market diversification and compensation for bearing diversified versus undiversified risk? Am I correct in this analysis or not? If not, what am I missing or doing wrong?

*Update*: Thanks to the commenters, I learned quite a lot about how to think about this as an abstract problem. I’d point to a Washington Post online article today that comments on a new study by the National Association for College Admission Counseling looking at the “arms race” in applications. I read the study fairly quickly, and in light of several of the comments below, was not sure that I agreed that it’s model reflects the incentives of the various players. In particular, it seemed to me to use a standard “arms race” gaming model that, for various reasons suggested in various of the comments, is not persuasively reflective of the incentives of the players.